Capacity Results for Finite-Field X-Channels with Delayed CSIT

01/09/2018
by   Alireza Vahid, et al.
0

We establish the capacity region of the two-user Finite-Field Fading X-Channel with delayed channel state information at the transmitters. We consider the most general case in which each transmitter has a common message for both receivers and a private message for each one of them. We derive a new set of outer-bounds for this problem that rely on an extremal entropy inequality. This inequality quantifies the ability of each transmitter in favoring one receiver over the other in terms of the delivered entropy when both receivers must obtain a baseline entropy. We show that the outer-bounds can be achieved by treating the X-Channel as a combination of a number of well-known problems such as the interference channel and the multicast channel. The capacity-achieving strategies of these sub-problems must be interleaved and carried on simultaneously in certain regimes in order to achieve the X-Channel outer-bounds.

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I Introduction

Alongside the two-user interference channel [1, 2, 3, 4], the two-user X-Channel is a canonical example to study the impact of interference in communication networks. In the X-Channel, each transmitter has a common message intended for both receivers as well as a private message intended for each receiver. This problem has been studied extensively in the literature and several interference management techniques have been proposed [5, 6, 7]. For instance under instantaneous channel state information (CSI) model, it was shown in [6] that interference alignment can provide a gain over baseline techniques (e.g.

, orthogonalization). This gain is expressed in terms of degrees-of-freedom (DoF) which captures the asymptotic behavior of the network normalized by the capacity of the point-to-point channel when power tends to infinity.

Attaining instantaneous channel state information at the transmitters (CSIT) in many real-world scenarios may not be feasible. In such cases, a more realistic model is the delayed CSIT in which by the time the CSI arrives at the transmitters, the channel has already changed to a new state. Under the delayed CSIT model, authors in [8] developed a scheme that achieves DoF. Later, it was shown that if we limit ourselves to linear encoding functions, then is indeed the optimal DoF [9]. These results provide valuable insight into the behavior of X-Channels. However, in information and communication theory, the ultimate goal is to understand the behavior of wireless networks for any signal-to-noise ratio (SNR). In other words, we are interested in capacity results rather than DoF-type results. Moreover, while one might argue that most practical communication protocols are linear, limiting the encoding functions to be linear removes the majority of potential encoding functions and from an information-theoretic perspective, this is not desirable. Finally, authors in [8] and [9] study a subset of X-Channels in which transmitters only have private messages for the receivers and the issue of common messages in X-Channels is not addressed, and as we will show, including common messages introduces new challenges.

In this work, we address these issues by deriving the capacity region of X-Channels with delayed CSIT and common messages under a finite-field fading model introduced in [10]

. In this model, the channel gains at each time are drawn from the binary field according to some Bernoulli distribution. The input-output relation of this channel model at time

is given by

(1)

where , the channel gains are in the binary field, is the transmit signal of transmitter at time , and is the observation of receiver at time . All algebraic operations are in . In the delayed CSIT model, we assume that the transmitters at time have access to

(2)

The X-Channel poses several new challenges when compared to the interference channel. In the interference channel, each transmitter has a private message for its corresponding receiver and to maximize the overall achievable rate, each transmitter tries to minimize the interference subspace at the unintended receiver. In the context of X-Channels, however, each transmitter has a private message for each one of the receivers which changes the interference dynamics of the problem since receivers are now interested in signals of both transmitters. On top of this, each transmitter has to deliver a common message to the receivers. In this work, we establish the capacity region of finite-field fading X-Channels under the delayed CSIT assumption. We derive a new set of outer-bounds for this problem. We also propose a distributed transmission strategy that harvests the delayed CSI to combine and to repackage previously communicated signals in order to deliver them efficiently. We show that this transmission strategy matches the outer-bounds, thus, characterizing the capacity region.

To derive the outer-bounds, we rely on an extremal entropy inequality to capture both the impact of delayed CSIT and the challenge of delivering the common messages. This inequality quantifies the ability of a transmitter to favor one receiver over the other in terms of provided entropy when: both receivers need to obtain some common entropy, and the transmitter has access to the delayed channel state information. In particular, this extremal inequality quantifies such that the following inequality holds for any input distribution:

(3)

where indicates the common message, and is the private message for . Using (3) and a genie-aided argument, we obtain the outer-bounds.

To achieve the outer-bounds, we treat the X-Channel as a combination of a number of well-known problems for which the capacity region is known. By adjusting different rates for the X-Channel, we can recover several other problems such as the interference channel, the multicast channel, the broadcast channel, and the multiple-access channel. We demonstrate how to utilize the capacity-achieving strategies of such problems in a systematic way in order to achieve the capacity region of the X-Channel. We show that, however, if we treat the X-Channel as a number of disjoint sub-problems, we will not achieve the capacity and in some regimes, we need to interleave the capacity-achieving strategies of different sub-problems and execute them simultaneously.

The rest of the paper is organized as follows. In Section II we formulate the problem. In Section III we present our main results and provide some insights. Sections IV and V are dedicated to the proof of the main results. Section VI concludes the paper.

Ii Problem Formulation

Fig. 1: Two-user Binary Fading X-Channel. All signals and the channel gains are in the binary field.

We consider the two-user Binary Fading X-Channel as illustrated in Fig. 1 with two transmitters and two receivers111In this work, we focus on the binary field. Extending the results from the binary field to larger Galois fields is possible and rather straightforward.. In the binary fading model, the channel gain from transmitter to receiver at time is denoted by , . The channel gains are either or (i.e.

), and they are distributed as independent Bernoulli random variables (independent across time and space). We consider the homogeneous setting in which

(4)

for , and we define .

At each time , the transmit signal of is denoted by , and the received signal at is given by

(5)

where the summation is in , and .

We define the channel state information (CSI) at time to be the quadruple

(6)

and for natural number , we set

(7)

where is defined in (6). Finally, we set

(8)

In this work, we consider the delayed CSIT model in which at time each transmitter has the knowledge of the channel state information up to the previous time instant (i.e. ) and the distribution from which the channel gains are drawn (i.e. ), . Since receivers only decode the messages at the end of the communication block, without loss of generality, we assume that the receivers have instantaneous knowledge of the CSI.

For the X-Channel, we consider the scenario in which , , wishes to reliably communicate

  1. message to both receivers,

  2. message to ,

  3. and message to ,

during uses of the channel. We assume that the messages and the channel gains are mutually independent and the messages are chosen uniformly at random.

For transmitter , let messages and be encoded as using the encoding function , which depends on the available CSI at , see Fig. 1. Receiver is interested in decoding and given by

(9)

and it will decode the messages using the decoding function :

(10)

An error occurs when

(11)

The average probability of decoding error is given by

(12)

where the expectation is taken with respect to the random choice of messages.

We define

(13)

A rate tuple is said to be achievable, if there exists encoding and decoding functions at the transmitters and the receivers respectively, such that the decoding error probabilities go to zero as goes to infinity. The capacity region is the closure of all achievable rate tuples.

Iii Main Results

In this section, we present the capacity region of the two-user Binary Fading X-Channel under the delayed CSIT assumption, and we provide some technical insights and interpretations of the main results.

Iii-a Statement of the Main Results

The following theorem establishes the capacity region of the two-user Binary Fading X-Channel with private and common messages under the delayed CSIT assumption.

Theorem 1.

The capacity region, , of the two-user Binary Fading X-Channel with private and common messages under delayed CSIT assumption as described in Section II is the set of all non-negative rates satisfying:

(14)

for , defined in (II), and

(15)

The capacity region is described by two sets of outer-bounds. The first set is referred to as the Broadcast Channel (BC) bounds. These bounds describe the capacity region of the broadcast channel formed by one of the transmitters and the two receivers assuming the other transmitter is eliminated. These bounds can be thought of as the generalization of the results in [11, 12, 13] for the two-user case to include common messages. The second set is referred to as the X-Channel (XC) bounds. We note that adding the two BC bounds results in

(16)

which is dominated by the XC bounds since for any , we have

(17)

The derivation of the outer-bounds relies on an extremal entropy inequality that quantifies the ability of each transmitter in favoring one receiver over the other in terms of the available entropy subject to two constraints: both receivers need to obtain a baseline entropy (to capture the common messages), and transmitters have access to the delayed CSI. This inequality characterizes the limit to which the unwanted subspace at one receiver can be scaled down while the desired subspace at the other receiver is maximized. We use this inequality and a genie-aided argument to derive the new outer-bounds.

The two-user X-Channel can be thought of as a generalization and a combination of several well-known problems. For instance, if and are the only non-zero rates, then the problem is equivalent to the multiple-access channel formed at , and if and are the only non-zero rates, then the problem is equivalent to the broadcast channel formed by . We demonstrate how to utilize the capacity-achieving strategies of other problems, such as the interference channel and the multicast channel, in a systematic way in order to achieve the capacity region of the X-Channel. We show that, however, if we treat the X-Channel as a number of disjoint sub-problems, we will not achieve the capacity in some regimes. In fact, in such regimes, we need to interleave the capacity-achieving strategies of different sub-problems and execute them simultaneously.

Fig. 2: The two-dimensional region of for . As increases, the symmetric capacity, , decreases.

Iii-B Illustration of the Main Results

To illustrate the results of Theorem 1, we focus on the symmetric case for , i.e.

(18)

Fig. 2 depicts the two-dimensional region of for . In this figure for a given , we define

(19)

An interesting observation is that the sum of and (i.e. ) determines the size of the region rather than the individual values. For instance, Fig. 2(c) is the same for , , and . Moreover, as increases, the symmetric capacity, , decreases. The reason is that providing more common entropy to the receivers reduces the ability of each transmitter in favoring one over the other in terms of available entropy. In other words, providing more common entropy to the receivers reduces transmitters’ ability in performing interference alignment.

Fig. 3: The capacity region of the two-user Binary Fading Interference Channel with delayed CSIT (shaded region in the figure) is included in the capacity region of the two-user Binary Fading X-Channel with delayed CSIT and .

Iii-C Comparison to the Interference Channel

For the two-user Binary Fading Interference Channel [10], there is no common message (i.e. ), and each transmitter only has a message for one receiver (i.e. ). Fig. 3 depicts the capacity region of the X-Channel for which includes the capacity region of the interference channel. We note that in the X-Channel, individual rates are limited by the capacity of the multiple-access channel (MAC) formed at each receiver (i.e. ), whereas in the interference channel the limit is the capacity of the point-to-point channel (i.e. ). Moreover, for some values of , the symmetric capacity of the X-Channel is strictly larger than that of the interference channel. This issue is further discussed in Section V-C and Fig. 8.

Iii-D The Broadcast Channel Bounds

So far, we focused on and and as a result, the BC bounds did not play a role. As mentioned earlier, the BC bounds describe the capacity region of the broadcast channel formed by one of the transmitters and the two receivers assuming the other transmitter is eliminated. Suppose we set and equal to (i.e. eliminating the second transmitter), and we focus on and . These rates correspond to and are governed by the BC bounds as depicted in Fig. 4.

Fig. 4: The BC bounds govern the behavior of the rates associated with each transmitter ( in this figure).

Iv Converse Proof of Theorem 1

In this section, we provide the converse proof of Theorem 1. The proof of the BC bounds has some similarities to that of the XC bounds and when possible we omit the duplications in the proofs.

BC Bounds: We first derive the Broadcast Channel bounds, i.e.

(20)

By symmetry, it suffices to prove (20) for . As mentioned before, this bound corresponds to the Broadcast Channel formed by when is eliminated. In our proof, this fact is captured by conditioning on and . We have

(21)

where as ; follows from the independence of messages; follows from Fano’s inequality; holds since messages are independent of channel realizations; follows from Claim 1 below; follows the fact that is a function of ; holds since . Dividing both sides by and letting , we get

(22)

Similarly, we can obtain

(23)
Claim 1.

For the two-user Binary Fading X-Channel with private and common messages under delayed CSIT assumption as described in Section II, we have

(24)

The proof of Claim 1 follows step by step that of Claim 2 that we will present and prove below and thus omitted.

XC Bounds: As mentioned before, the XC bounds cannot be obtained from the BC bounds. However, the derivation resembles the one we provided for the BC bounds with some modifications. To derive the XC bounds, we have

(25)

where as ; follows from the independence of messages; follows from Fano’s inequality; holds since messages are independent of channel realizations; follows from Claim 2 below; holds since

(26)

Dividing both sides by and letting , we get

(27)

Similarly, we can obtain

(28)
Claim 2.

For the two-user Binary Fading X-Channel with private and common messages under delayed CSIT assumption as described in Section II, we have

(29)
Proof.

We first note that

(30)

Thus, proving (29) is equivalent to proving

(31)

We have

(32)

where follows from the fact that all signals at time are independent of future channel realizations; holds since ; is true since transmit signal is independent of the channel realization at time ; holds since conditioning reduces entropy; holds since ; follows from the definition of ; is true since all signals at time are independent of future channel realizations;

follows from the chain rule and the non-negativity of the entropy function for discrete random variables. ∎

This completes the converse proof of Theorem 1, and in the following section we present the achievability proof.

V Achievability Proof of Theorem 1

X-Channels can be thought of as a generalization of several known problems such as interference channels, broadcast channels, multiple-access channels, and multicast channels. In the previous section, we developed a set of new outer-bounds for this problem. In this section, we show that a careful combination of the capacity-achieving strategies for other known known problems will achieve the capacity region of the X-Channel. However, in Section V-C we show that if we treat the X-Channel as a number of disjoint sub-problems, we will not achieve the capacity and in some regimes, we need to interleave the capacity-achieving strategies of different sub-problems and execute them simultaneously.

To describe the transmission strategy, we first present two examples. The first example describes a symmetric scenario associated with Fig. 2(b) and the second example describes a scenario in which transmitters achieve unequal rates. After the examples, we present the general scheme.

Fig. 5: (a) The two-user multicast channel and its capacity region, the capacity can be achieved without using the delayed CSI; (b) The two-user binary fading interference channel and its capacity region; (c) The two-user binary fading interference channel with swapped IDs and its capacity region.

V-a Example 1: Symmetric Sum-Rate of Fig. 2(b)

Suppose for , we wish to achieve the sum-capacity of with

(33)

as shown in Fig. 2(b).

For this particular example, we treat the X-Channel as three separate problems listed below at different times, and we show that this strategy achieves the capacity.

  • For the first third of the communication block, we treat the X-Channel as a two-user multicast channel as depicted in Fig. 5(a) in which each transmitter has a message for both receivers. For the two-user multicast channel with fading parameter , the capacity region matches that of the multiple-access channel formed at each receiver [10] and depicted in Fig. 5(a) as well.

  • For the second third of the communication block, we treat the X-Channel as a two-user interference channel in which wishes to communicated with , see Fig. 5(b). The capacity region of this problem is given in [10] and depicted in Fig. 5(b).

  • During the final third of the communication block, we treat the X-Channel as a two-user interference channel with swapped IDs in which wishes to communicated with , see Fig. 5(c). In the homogeneous setting of this work, the capacity region of this interference channel with swapped IDs matches that of the previous case and is depicted in Fig. 5(c).

Achievable Rates: We note that as the communication block length, , goes to infinity, so do the communication block lengths for each sub-problem. Thus, during the first third of the communication block, we can achieve symmetric common rates arbitrary close to . Normalizing to the total communication block, we achieve which matches the requirements of (V-A). From [10] we know that for the two-user binary fading interference channel with delayed CSIT and , we can achieve symmetric rates of . Normalizing to the total communication block, we achieve which matches the requirements of (V-A). Finally, during the final third of the communication block we treat the problem as a two-user interference channel with swapped IDs in which we can achieve symmetric rates of . Normalizing to the total communication block, we achieve which again matches the requirements of (V-A). Thus, with splitting up the X-Channel into a combination of three known sub-problems, we can achieve the capacity region described in Theorem 1.

Fig. 6: (a) The two-user broadcast channel with a single common message; (b) The two-user broadcast channel with delayed CSIT and private messages, and its capacity region.

V-B Example 2: Unequal Rates

In the previous subsection we focused on a symmetric setting. Here, we discuss a scenario in which transmitters have different types of messages with different rates for each receiver. More precisely, we consider the region in Fig. 2(c) for , and

(34)

In this case, we can think of the X-Channel in this case as two sub-problems that coexist at the same time as described below.

  • The Binary Fading Broadcast Channel from as in Fig. 6(a) in which a single message is intended for both receivers. For this problem, the capacity can be achieved using a point-to-point erasure code of rate .

  • The Binary Fading Broadcast Channel from as in Fig. 6(b) with delayed CSIT in which the transmitter has a private message for each receiver. For this problem, the capacity region is given in [11, 12] and depicted Fig. 6(b). As described below, in order to be able to decode the messages in the presence of the broadcast channel from , we first encode and using point-to-point erasure codes of rate , and treat the resulting codes as the input messages to the broadcast channel of Fig. 6(b).

Achievable Rates: At each receiver the received signal from is corrupted (erased) half of times by the signal from . As a result, when we implement the capacity-achieving strategy of [11, 12], we only deliver half of the bits intended for each receiver. However, since we first encode and using point-to-point erasure codes of rate , obtaining half of the bits is sufficient for reliable decoding of and . Thus, we achieve

(35)

which again matches the requirements of (V-B). At the end of the communication block, receivers decode and , and remove the contribution of from their received signals. After removing the contribution of , the problem is identical to the broadcast channel from as in Fig. 6(a) for which we can achieve a common rate of .

Fig. 7: (a) The two-user multicast channel and its capacity region, the capacity can be achieved without using the delayed CSI; (b) The two-user binary fading interference channel and its capacity region; (c) The two-user binary fading interference channel with swapped IDs and its capacity region.

V-C Transmission Strategy

The two examples we have provided so far demonstrate the key ideas behind the transmission strategy and dividing the problem into a number of known sub-problems. Suppose we would like to achieve

(36)
Fig. 8: The sum-capacity of the X-Channel vs. that of the Interference Channel. To have a fair comparison, in the X-Channel we set .

An important difference between the X-Channel and the interference channel is the fact that in the latter scenario, the individual rates are limited by the capacity of a point-to-point channel, i,e, . As a result, for the interference channel we have [10]:

(37)

However, in the X-Channel no such limitation exists, and we have

(38)

The difference is depicted in Fig. 8 for . This means that if we naively try to use the capacity-achieving strategies of the sub-problems independently, we cannot achieve the capacity region of the X-Channel. The key idea to over come this challenge is to take an approach similar to the one we presented in Section V-B and run different strategies simultaneously as described below.

The strategy that achieves the rates in (V-C) is similar to what we presented in Section V-A. Define

(39)

First suppose

(40)

Then the transmission strategy is as follows.

  • For the first fraction of the communication block, we treat the X-Channel as a two-user multicast channel as depicted in Fig. 7(a) in which each transmitter has a message for both receivers. For the two-user multicast channel with fading parameter , the capacity region matches that of the multiple-access channel formed at each receiver [10] and depicted in Fig. 7(a) as well.

  • During a fraction of the communication block, we treat the X-Channel as a two-user interference channel in which wishes to communicated with , see Fig. 7(b). An instance of the capacity region of this problem is given in [10] and depicted in Fig. 7(b).

  • During a fraction of the communication block, we treat the X-Channel as a two-user interference channel with swapped IDs in which wishes to communicated with , see Fig. 7(c). In the homogeneous setting of this work, the capacity region of this interference channel with swapped IDs matches that of the previous case and an instance of it is depicted in Fig. 7(c).

Achievable Rates: With this strategy fraction of the times, we achieve a common rate, , of , while fraction of the times, we achieve individual rates

(41)

The overall achievable rate matches in this case. Now, consider the case in which

(42)

We need to modify the strategy slightly. The transmission strategy for the interference channel consists of two phases. During Phase 1, uncoded bits intended for different receivers are transmitted. During Phases 2, using the delayed CSIT, the previously transmitted bits are combined and repackaged to create bits of common interest. These bits are then transmitted using the capacity-achieving strategy of the multicast problem. In the modified strategy for the X-Channel, Phase 1 consists of two sub-phases. In the first sub-phase, both transmitters send out bits intended for while in the second sub-phases, bits intended for are communicated. This way we take full advantage of the entire signal space at each receiver and the individual rates are no longer limited by the capacity of a point-to-point channel. The second phase is identical to the Interference Channel.

Fig. 9: The multiple-access channel formed at and its capacity region.

Achieving unequal rates for different users follows the same logic with careful application of sub-problems as described in Section V-B. Another example is corner point

(43)

of Fig. 2(a). This corner point corresponds to the Multiple-Access Channel formed at as depicted in Fig. 9.

In this section, we presented the transmission strategy for where . We also discussed and highlighted the strategy for other corner points of the capacity region.

Vi Conclusion

We established the capacity region of finite-field fading X-Channels with common messages and delayed CSIT. We presented a new set of outer-bounds for this problem that relied on an extremal entropy inequality developed specifically for this problem. We then showed how the outer-bounds can be achieved by treating the X-Channel as a combination of a number of well-known problems such the interference channel and the multicast channel.

An important future work is to study Gaussian X-Channels with delayed CSIT. One approach could be to extend our results to the multi-layer finite-field fading setting similar to [14] and then, derive the capacity region of the Gaussian X-Channels to within a constant number of bits.

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